Question

If the tangent at any point of the curve $x^{2/3}+y^{2/3}=a^{2/3}$ meets the axis of co-ordinates in A and B, then the locus of mid-point of AB is a ..............

• A. straight line
• B. ellipse
• C. circle
• D. parabola

Answer 1

The correct option is C circle

General parametric form of tangent: $x\cos \theta +y\sin \theta =a\sin \theta \cos \theta$ $\Rightarrow \frac{x}{\sin \theta }+\frac{y}{\cos \theta }$
$\therefore A(a^{\frac{2}{3}}{x}^{\frac{1}{3}},0),B(0,a^{\frac{2}{3}}{y}^{\frac{1}{3}})$ as $A(a\sin \theta ,0),B(0,a\cos \theta )$
If $(h,k)$ be mid points of $AB$ then $2h=a^{\frac{2}{3}}{x}^{\frac{1}{3}},2k=a^{\frac{2}{3}}{y}^{\frac{1}{3}}$
Squaring and adding $4(h^2+k^2)=a^{\frac{4}{3}}(x^{\frac{2}{3}}+y^{\frac{2}{3}})=a^{\frac{2}{3}}{a}^{\frac{2}{3}}=a^2$
Locus is $x^2+y^2=\frac{a^2}{4}$, which is a circle.
In parametric form,
$2h=a\sin \theta ,2k=a\cos \theta$
$\therefore 4(h^2+k^2)=a^2.1$
$\therefore$ Locus is $x^2+y^2=\frac{a^2}{4}$, which is a circle.